Fall 2008 Stephen

In many ways my current research efforts were shaped by my experiences at my undergraduate institution, the Rose-Hulman Institute of Technology. Rose-Hulman is a small, primarily undergraduate, engineering school in rural Indiana. As a result, the mathematics department offerings tended towards the eclectic with courses in the traditional core of research mathematics rarely offered. These courses were instead replaced by a motley collection of theoretical and applied courses which would appeal to the engineering student body. As a result of Rose’s engineering focus, in addition to the rather unusual collection of courses which I took as an undergrad, there was also a philosophical viewpoint towards problem solving I absorbed in my time at Rose. Because of this focus, there was an emphasis on group/collaborative work throughout the curriculum. Even on assignments that were ostensibly individual, the expectation was that although all the work would be your own, you would have discussed/consulted with your fellow students. The range of mathematics courses at Rose led me naturally to the program in Algorithms, Combinatorics, and Optimization (ACO) at the Georgia Institute of Technology. The ACO program is an interdisciplinary degree-granting program offered jointly by the School of Mathematics, the School of Computer Scinece, and the H. Milton Stewart School of Industrial and Systems Engineering. However, the interdisciplinary nature of the program did pose some problems for me, as I had too many options for potential advisors and thus had a hard time settling on one. Ultimately, it was a chance encounter with Prof. Ed Scheinerman of Johns Hopkins University that led me to my thesis topic and finalized my choice of advisor. Prof. Scheinerman described to me his recent work with Kraetzl and Nickel on the random dot product graph model [19]. In this model, they considered a random graph formed by assigning to every vertex a single value, X, distributed on [0, 1] such that P (X ≤ t) = t/α for some α > 0. Each edge was then present independently with probability equal to the product of the endpoints. They were able to show that this model exhibits positive clustering, has a power-law-like degree distribution for large degrees, and the giant component has diameter at most 5. In describing this model, Prof. Scheinerman brought to my attention several extensions of the model whose behavior he wished to resolve. Specifically, can X be replaced by a vector and multiplication be replaced by the dot product; can the model be modified so as to be directed; and can the model be generalized to a sparse(r) model? In response to Prof. Scheinerman’s questions, independently and in collaboration with my advisor Prof. Milena Mihail and Prof. Scheinerman, I have developed answers to these questions. Specifically, we generalized the model to the following: Let μ be a distribution on Rd satisfying mild technical conditions and let g(n) be a monotonically increasing function. Then to each vertex v assign a vector Xv distributed according to μ. Each edge {u, v} is then present independently with probability 〈Xu,Xv〉/g(n). With this generalized model, we were able to show that unless μ is concentrated on a single point, the resulting graph exhibits positive clustering. By proving a new concentration result for binomial random variables with random parameters, we were also able to describe the relationship between μ and the log-histogram of the degree distribution. We also characterized the diameter of the model by reducing the problem to the asymptotic behavior of a class of Erdős-Rényi models. To our knowledge, this is the first diameter result that has proceeded via a reduction rather than through a direct calculation. Combining these results with the natural semantic interpretation of the vectors assigned to vertices, we feel that this model could be a significant advance in the study of complex networks and their models.